## Analysis and LogicThis volume comprises articles from four outstanding researchers who work at the cusp of analysis and logic. The emphasis is on active research topics; many results are presented that have not been published before and open problems are formulated. Considerable effort has been made by the authors to integrate their articles and make them accessible to mathematicians new to the area. |

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### Contents

Introduction | 3 |

Normed Space Structures | 10 |

Signatures | 13 |

Ultrapowers of Normed Space Structures | 17 |

Positive Bounded Formulas | 21 |

Basic Model Theory I | 31 |

QuantifierFree Formulas | 34 |

Ultraproducts of Normed Space Structures | 38 |

Actions of the Infinite Symmetric Group | 133 |

Turbulence I Overview | 137 |

Turbulence II Basic Facts | 143 |

Turbulence III Induced Actions | 146 |

Turbulence IV Some Examples | 150 |

Turbulence V Calmness | 152 |

Turbulence VI The First Main Theorem | 157 |

Turbulence VII The Second Main Theorem | 163 |

Basic Model Theory II | 43 |

Isomorphic Ultrapowers | 55 |

Alternative Formulations of the Theory | 72 |

Homogeneous Structures | 75 |

More Model Theory | 79 |

Types | 96 |

References | 107 |

Index of Notation | 111 |

112 | |

Actions of Polish Groups and Classification Problems | 115 |

Introduction | 117 |

The General GlimmEffros Dichotomy | 120 |

Actions of Polish Groups | 123 |

Actions of Countable Groups | 126 |

Actions of Locally Compact Groups | 131 |

References | 183 |

Index | 186 |

On Subspaces Asymptotic Structure and Distortion of Banach Spaces Connections with Logic | 189 |

Introduction | 191 |

Background material The 60s and 70s | 194 |

The unconditional basic sequence problem and connections with distortion | 208 |

Gowers dichotomy A block Ramsey Theorem | 217 |

Distortion | 223 |

Asymptotic structure | 233 |

Ordinal Indices | 244 |

The homogeneous Banach space problem | 252 |

Concluding remarks | 254 |

256 | |

265 | |

### Common terms and phrases

action algebras approximate elementary extension approximate model approximately satisfiable Banach spaces block subspace Borel equivalence relation closed subgroup comeager concretely classifiable consider construction Corollary countable Borel equivalence countable sets countable structures define Definition denote dense elementary extension elements of suitable embedding equivalence classes equivalent norm ergodic theory example exists finite subset following conditions function symbol given Hjorth 00 hyperfinite induction infinite dimensional invariant isomorphic Kechris Krivine L-formula linear logic Math model of F model theory nonstandard analysis nonstandard hull normed space structures Polish group Polish space positive bounded formulas positive bounded sentences positive bounded theory problem Proposition prove quantifier elimination Ramsey theorem rational number real number reflexive result set of positive signature spreading model substructure suitable sorts Suppose topology turbulent ultrafilter ultrapower ultraproducts unconditional basic sequence uniform class uniform continuity unit vector basis weakly null